ENSC 201 Assignment 6 Model Answers
6.1
A Placidian company needs to purchase ten oscilloscopes during the coming year. The oscilloscopes
are top-of-the-line models, with a built-in Pentium 5 processor, and cost $80 000 each. The company is
presenting an argument to the Internal Revenue Service (‘IRS’) that they should be considered as
computers, and depreciated, using straight-line depreciation, over a five-year period. The IRS thinks
that they are generic electronic equipment, which is depreciated, again using straight-line depreciation,
over a seven-year period. The company pays tax at 50%, and its pre-tax cash flow every year is at least
$500 000. Its after-tax MARR is 5%. What is the present value to the company of convincing the IRS
that they are computers?
If the oscilloscopes are depreciated over five years, you will deduct $160,000 a year from your pre-tax
cash flow every year for the next five years. This will save you $80,000 a year in taxes, and the present
worth of these tax savings is 80,000(P/A,0.05,5).
If the oscilloscopes are depreciated over seven years, you will deduct $114,000 a year from your pretax cash flow every year for the next seven years. This will save you $57,000 a year in taxes, and the
present worth of these tax savings is 57,000(P/A,0.05,7).
So the value of convincing the IRS that they’re computers is the difference between these two present
worths, which is (80,000(P/A,0.05,5) – 57,000(P/A,0.05,7))
= 80,000(4.329) – 57,000(5.786)
= $16,518
6.2
A large London printing company has a printing press with five years of life remaining to it, and its
function will be needed for exactly that period. It costs £19 000 per year to operate. Right now its
market value is £8 000, and in five years time its salvage value will be £1 000. A new press of
advanced design can perform the same function as the current machine for an operating cost of just
£12 000 per year. It will cost £24 000 to purchase, and will have a salvage value of £6 000 after five
years of use.
The company has a 52% effective tax rate, and its after-tax MARR is 20%. Both machines are
depreciated by declining balance depreciation at 25%. Should the machine be replaced?
The simplest way of approaching this question is to consider the difference in cash flows that results
from selling the old machine and buying the new one.
To dealing with the initial and salvage costs we calculate the tax benefit factor, or `TBF’ (see page 283
of the text; the TBF is 1-CTF, where the CTF is the concept discussed in Lecture 11.). The TBF is:
TBF = td/(i+d) = 0.52 × 0.25/(0.2+0.25) = 0.289
So if we sell the old machine after five years, the after-tax value of the price we receive is
1 000× (1 - 0.289) = £711. Similarly, if we buy the new machine and sell it for salvage after five years,
its after-tax value to the company at that time will be £4 267.
So the present cost of selling the old machine and buying the new one, together with the resultant
savings in operating costs, is:
PW
= (24 000-8 000)(1-TBF) - (6 000-1 000)(1-TBF)(P/F 0.2,5) - (19 000-12 000)(1-t)(P/A,0.2,5)
= 16 000×0.711 – 5 000×0.711×0.4019 – 7 000×0.48 ×2.9906
= 11 376 – 1 429 – 10 048
= -101
Buying the new machine leaves the company just £101 better off; in principle this implies the company
should buy it, but it might be more reasonable to say that financial analysis does not identify a clear
winner – the margins of error in our estimates of salvage price and operating costs are almost
certainly more than £101.
6.3
You operate a deep mine in South Africa, which will be productive for the next five years.
Groundwater seeps into the mine, and a powerful pump must be purchased to remove it.
The Mandela pump sells for R 14 000, costs R 6 000 a year to operate, and will have a salvage value of
R 5 000 in five years time. The Kruger pump sells for R 10 000, costs R 7 000 a year to operate, and
will have a salvage value of R 2 000 in five years time. A third alternative is to apply a waterproofing
treatment to the mine walls now. This will require the immediate expenditure of R 25 000, and
R 1 000 annual maintenance thereafter. If your pre-tax MARR is 20%, which option will you choose?
Repeat the study, this time with an after-tax analysis, assuming that your tax rate is 50%, the pumps are
depreciable at 20% per annum, and your after-tax MARR is 10%. The initial cost of waterproofing is
deductible from your pre-tax income for the current year. Does your conclusion change?
The pre-tax version of this problem can be solved by application of the formulae we already know, but,
since the after-tax version is most conveniently solved by use of a spreadsheet, we shall use the same
method for both versions. We assume that all operating and maintenance costs are paid at the end of
the year in which they occur.
From spreadsheet 6.4a(PreTax).xls, we see that the waterproofing option has the lowest present cost,
and should therefore be chosen. We then repeat the study for the after-tax case (spreadsheet
6.4b(AfterTax).xls). The waterproofing option still looks like the best – the fact that we can deduct the
cost of the waterproofing treatment from the tax bill in the first year increases its advantage.
6.4
You run a CAD/CAM consultancy in South Ontario. You employ ten analysts, each on an annual
salary of $100 000, and have an ageing collection of ten workstations of various kinds. You are
considering four possible strategies: replace all the workstations with new, low-end workstations
costing $50 000 each; rent ten low-end workstations at $20 000 per year each; buy five high-end
workstations at $150 000; or carry on with your current equipment, in which case you can expect an
annual total repair bill of $100 000, going up by 20% per year. If you buy the high-end workstations,
you can retire one of your analysts at once. The same four options are available to you every year over
the next five years; after that, the technology will have changed too much to make any useful
predictions. Your current workstations have zero salvage value; the cost of the low-end workstations
will drop by 40% per year, while the high-end workstations cost drops at 30% per year. Rental costs
stay the same from year to year. Your company is taxed at 50%, and has an after-tax MARR of 10%.
Perform two analyses to determine the best strategy: first, calculate an appropriate pre-tax MARR and
perform a pre-tax analysis. Secondly, use the after-tax MARR and perform an after-tax analysis. Does
your conclusion change?
Note that the market value of the computers may not match the depreciation rate specified by Canadian
tax regulations. Canada tax law says that they’re in Asset Class 10, which means that you can’t
depreciate them by more than 30% per year. You can assume that salaries, repairs and rental costs can
be deducted from your pre-tax cash flows in the year in which they occur.
Pre-tax analysis
Consider a study period of five years. An appropriate value for your pre-tax MARR would be 20%.
Suppose that the salvage value of the workstations drops at the same rate as their cost. The present
cost of keeping your current equipment another N years is
$100 000 (P/A,0.2,0.2,N)
The present cost of bringing in low-end workstations after N years is
$500 000((0.6N)(P/F,0.2,N) - (0.6 5)(P/F,0.2,5))
The present cost of bringing in high-end workstations after N years is
$750 000((0.7N)(P/F,0.2,N) - (0.7 5)(P/F,0.2,5)) – 100 000(P/A,0.2,(5-N))(P/F,0.2,N)
The present cost of starting to rent workstations after N years is
$200 000(P/A,0.2,(5-N))(P/F,0.2,N)
The present value of keeping your current equipment N years, then switching to one of the three
alternatives, is calculated in the accompanying spreadsheet, 6.5(PreTax).xls. All figures on the
spreadsheet are in hundreds of dollars. Comparing the last three columns, we see that the best
strategy is to wait two years, then replace your old machines with high-end workstations.
N
0
1
2
3
4
5
(P/F,0.2,N)
1.0000
0.8333
0.6944
0.5787
0.4823
0.4019
1000(P/A,0.2,0.2,N)
0
833
1 739
2 609
3 478
4 348
(P/A,0.2,(5-N))
2.9906
2.5887
2.1065
1.5278
0.8333
0.0000
Low-Cost
4 844
3 177
2 833
3 077
3 635
4 348
High Cost
4 003
2 544
2 322
2 707
3 438
4 348
Rent
5 981
5 148
4 665
4 377
4 282
4 348
After-tax analysis
Again consider a study period of five years. If we were doing a pre-tax analysis, the present cost of
keeping your current equipment another N years would be
$100 000 (P/A,0.2,0.2,N)
However, you can now deduct the cost of your repairs from your pre-tax cashflow, which effectively
halves their cost. Adjusting the MARR to its after-tax value of 0.1, the after-tax present cost is now:
$50 000 (P/A,0.2,0.1,N)
The conversion factor can be evaluated as
(P/A,0.2,0.1,N) = ((1+i0)N-1)/(i0(1+i0)N(1+g))
where i0 = (1+i)/(1+g) -1 = 1.1/1.2 – 1 = -0.083
The pre-tax present cost of bringing in low-end workstations after N years would be
$500 000((0.6N)(P/F,0.2,N) - (0.6 5)(P/F,0.2,5))
To find the after-tax cost, we adjust both the initial cost and the salvage revenue using the tax benefit
factor, or `TBF’ (see page 283 of the text; the TBF is 1-CTF, where CTF is the concept introduced in
Lecture 11). The TBF is:
TBF = td/(i+d) = 0.5 × 0.3/(0.1+0.3) = 0.375
And we reduce the income from our salvage revenue by a factor (1-TBF) to take into account the loss
of future depreciation tax benefits. When we correct the purchase cost, we have to take into account the
`half-year rule’; from Example 8.9 in the text, we note that application of this rule implies that the
effective reduction in purchase cost is given by the factor:
(1-TBF ×0.5 ×(1+(P/F,i,1))), which we shall call `1-TBF*’. In this case,
1-TBF* = 1-TBF ×0.5 ×(1+(P/F,0.1,1))
= 1 – 0.375 ×0.5 ×(1 + 0.9091)
= 1 – 0.358
= 0.642
So the after-tax cost of bringing in low-end workstations after N years is
$500 000((0.6N)(P/F,0.1,N)(1-TBF*) - (0.6 5)(P/F,0.1,5)(1-TBF))
Similarly, the pre-tax present cost of bringing in high-end workstations after N years would be:
$750 000((0.7N)(P/F,0.2,N) - (0.7 5)(P/F,0.2,5)) – 100 000(P/A,0.2,(5-N))(P/F,0.2,N)
We adjust the initial and salvage prices by the factors (1-TBF*) and (1-TBF) respectively, and note that
the benefit of the salaries saved is reduced by the factor (1-t), or 50%.
So the after-tax cost of bringing in high-end workstations after N years is
$750 000((0.7N)(P/F,0.2,N)(1-TBF*) - (0.7 5)(P/F,0.2,5) (1-TBF)) – 50 000(P/A,0.1,(5-N))(P/F,0.1,N)
Lastly, the pre-tax present cost of starting to rent workstations after N years would be
$200 000(P/A,0.2,(5-N))(P/F,0.2,N)
Since the rental cost can be deducted from our pre-tax cashflow, the after-tax cost of rental is reduced
by the factor (1-t). So the present value of the after-tax rental cost is:
$100 000(P/A,0.1,(5-N))(P/F,0.1,N)
The present value of keeping your current equipment N years, then switching to one of the three
alternatives, is calculated in the accompanying spreadsheet, 6.5(AfterTax).xls. All figures on the
spreadsheet are in thousands of dollars. Comparing the last three columns, we see that the best
strategy is still to wait two years, then replace your old machines with high-end workstations.
Years into
Future (N)
(P/F,0.1,
N)
50(P/A,0.2,0.1,
N)
0
1.0000
0.0
1
0.9091
45.6
2
0.8264
95.4
3
0.7513
149.7
4
0.6830
208.9
5
0.6209
273.5
0.6^N
1.000
0
0.600
0
0.360
0
0.216
0
0.129
6
0.077
8
0.7^N
1.000
0
0.700
0
0.490
0
0.343
0
0.240
1
0.168
1
(P/A,0.1,(
5-N))
Buy LowCost
Buy High
Cost
Ren
t
3.7908
306
242
379
3.1699
206
158
334
2.4869
176
138
301
1.7355
187
159
280
0.9091
222
207
271
0.0000
274
274
273